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5-47E

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Statistics For Business And Economics

Found in: Page 321

Book edition 13th

Author(s) James T. McClave, P. George Benson, Terry Sincich

Pages 888 pages

ISBN 9780134506593

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Short Answer

Question: Hotel guest satisfaction. Refer to the results of the 2015 North American Hotel Guest Satisfaction Index Study, Exercise 4.49 (p. 239). Recall that 15% of hotel guests were “delighted” with their experience (giving a rating of 10 out of 10); of these guests, 80% stated they would “definitely” recommend the hotel. In a random sample of 100 hotel guests, find the probability that fewer than 10 were delighted with their stay and would recommend the hotel.

The probability that fewer than 10 () delighted with their stay and would recommend the hotel is 0.2676.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Given information

Let p denotes the proportion of guests who were delighted with their stay and would recommend the hotel.

15% of guests were delighted with their experience, and of these, 80% would recommend the hotel

Therefore,

.

A random sample of size 100 is selected.

Computing the required probability

The probability that fewer than 10 () delighted with their stay and would recommend the hotel obtain as follows

P (\hat{p} < . 10) = P (\frac{\hat{p} - p}{σ_{\hat{p}}} < \frac{0.10 - p}{σ_{\hat{p}}}) = P (Z < \frac{0.10 - 0.12}{\sqrt{\frac{0.12 \times 0.88}{100}}}) = P (Z < \frac{- 0.02}{\sqrt{0.001056}}) = P (Z < \frac{- 0.02}{0.0324}) = P (Z < - 0.62) = 0.2676

In the z-table, the value at the intersection of -0.60 and 0.02 is the required probability.

Therefore, the required probability is 0.2676.

Most popular questions for Math Textbooks

Rental car fleet evaluation. National Car Rental Systems, Inc., commissioned the U.S. Automobile Club (USAC) to conduct a survey of the general condition of the cars rented to the public by Hertz, Avis, National, and Budget Rent-a-Car.* USAC officials evaluate each company’s cars using a demerit point system. Each car starts with a perfect score of 0 points and incurs demerit points for each discrepancy noted by the inspectors. One measure of the overall condition of a company’s cars is the mean of all scores received by the company (i.e., the company’s fleet mean score). To estimate the fleet mean score of each rental car company, 10 major airports were randomly selected, and 10 cars from each company were randomly rented for inspection from each airport by USAC officials (i.e., a sample of size n = 100 cars from each company’s fleet was drawn and inspected).

a. Describe the sampling distribution of x, the mean score of a sample of n = 100 rental cars.

b. Interpret the mean of x in the context of this problem.

c. Assume $μ = 30$ and $σ = 60$ for one rental car company. For this company, find $P (\bar{x} ⩾ 45)$ .

d. Refer to part c. The company claims that their true fleet mean score “couldn’t possibly be as high as 30.” The sample mean score tabulated by USAC for this company was 45. Does this result tend to support or refute the claim? Explain.

Consider the following probability distribution:

a. Find $μ$ .

b. For a random sample of n = 3 observations from this distribution, find the sampling distribution of the sample mean.

c. Find the sampling distribution of the median of a sample of n = 3 observations from this population.

d. Refer to parts b and c, and show that both the mean and median are unbiased estimators of $μ$ for this population.

e. Find the variances of the sampling distributions of the sample mean and the sample median.

f. Which estimator would you use to estimate $μ$ ? Why?

Question: Consider the following probability distribution:

a. Find $μ$ and $σ^{2}$ .

b. Find the sampling distribution of the sample mean x for a random sample of n = 2 measurements from this distribution

c. Show that $x$ is an unbiased estimator of $μ$ . [Hint: Show that $\underset{}{\sum (x)} = \underset{}{\sum x p (x)} = μ$ . ]

d. Find the sampling distribution of the sample variance $s^{2}$ for a random sample of n = 2 measurements from this distribution.

Consider the following probability distribution:

Find and $σ^{2}$ .
Find the sampling distribution of the sample mean x for a random sample of n = 2 measurements from this distribution
Show that $x$ is an unbiased estimator of $μ$ . [Hint: Show that.] $\sum_{}^{} (x) = \sum_{}^{} x p (x) = μ$ .]
Find the sampling distribution of the sample variance $s^{2}$ for a random sample of n = 2 measurements from this distribution.

Question: Refer to Exercise 5.3.

Show that $x$ is an unbiased estimator of $μ$ .
Find $σ_{x}^{2}$ .
Find the x probability that x will fall within $2 σ_{x}$ of $μ$ .

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